2.1. Book value return on assets

Let $C_t$ denote the amount of cash held by the company at time t. Let further $\mathcal{A}_t$ denote the set of assets, other than the cash account, in the company’s portfolio at time t. Each asset, $a\in \mathcal{A}_t$ , has at each time step a book value, $BV_t^a$ , and a market value, $MV_t^a$ . The difference is the unrealized gains or losses, $UG_t^a = MV_t^a - BV_t^a$ . The total portfolio values are correspondingly

\[ BV_t = \sum_{a\in\mathcal{A}_t}BV_t^a + C_t,\quad MV_t = \sum_{a\in\mathcal{A}_t}MV_t^a + C_t,\quad UG_t = MV_t - BV_t.\]

In accordance with Solvency II regulation, we assume that all market values are determined in an arbitrage-free manner. Concretely, we assume an interest rate model specified on a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{Q})$ such that $\mathbb{Q}$ is the risk neutral measure for the money market numeraire $B_t = \Pi_{j=0}^{t-1}(1+F_j)$ , where $F_{t}$ is the simple one year forward rate valid between t and $t+1$ . For example, this setting is satisfied for the LIBOR market model with respect to the spot measure (Brigo and Mercurio, Reference Brigo and Mercurio2006). Further, all stochastic processes shall be adapted to $(\mathcal{F}_t)$ , and all expected values, $E[\;\cdot\;]$ , are with respect to $\mathbb{Q}$ . The discount factor from s to $t< s$ is given by $ D(t,s) = \Pi_{j=t}^{s-1}(1+F_j)^{-1} = B_t B_s^{-1}$ . Hence, the value of a zero coupon bond paying one unit of currency at s is given by $P(0,s) = E[B_s^{-1}]$ , and more generally, we have $P(t,s) = E[D(t,s)]$ for $t< s$ .

Let $a\in\mathcal{A}_{t-1}$ and let $cf_s^a$ be the cash flow generated by a at $s\ge t$ . For example, if a is a bond, stock equity position or real estate investment, then $cf_t^a$ would correspond to a coupon or principle payment, dividend yield or rental revenue, respectively. The cash flow, $cf_s^a$ , goes to the cash account, $C_s$ , and increases it accordingly. Cash flows can be influenced by market movements (affecting, e.g., dividend payments) or by management decisions. Indeed, the management may decide at any time, $s<T_a$ , prior to the asset’s maturity to sell the asset and realize the market value $MV_s^a$ as a cash flow. This step is called realization because it converts the unrealized gains $UG_s^a$ into a book value return. If an asset, a, is sold at $s<T_a$ then its book value is terminated, $BV_s^a=0$ , and its market value is converted to a cash flow, $cf_s^a = (cf_s^a)' + MV_a^s$ , where $(cf_s^a)'$ are those cash flows (e.g., coupon payments or dividend yield) that result from holding a over the period $[s-1,s]$ .

The cash flow process, $cf_t^a$ , is an $(\mathcal{F}_t)$ -adapted process.

To avoid notational difficulties, we set $cf_s^a=0 = MV_s^a$ for $s>T_a$ . At $t-1$ the forward rate $F_{t-1}$ is known, that is it is $(\mathcal{F}_{t-1})$ -adapted. We have the relation

(2.1) \begin{align} MV_{t-1}^a &= E\left[\sum_{s=t}^{T}D(t-1,s)cf_s^a\Big| \mathcal{F}_{t-1}\right] + E\Big[D(t-1,T)MV_T^a\Big| \mathcal{F}_{t-1}\Big] \nonumber \\ & = \Big(1+F_{t-1}\Big)^{-1} \Big(E\Big[cf_t^a\Big|\mathcal{F}_{t-1}\Big] + E\Big[MV_t^a\Big| \mathcal{F}_{t-1}\Big]\Big) \end{align}

where the first line involves the final market value, $MV_T^a$ , at the end of the projection which satisfies $MV_T^a = 0$ if $T_a\le T$ .

The book value return, $ROA_t^a$ , at t of a single asset $a\in\mathcal{A}_{t-1}$ is

(2.2) \begin{equation} ROA_t^a = cf_t^a + \Delta BV_t^a,\end{equation}

and that of the portfolio is

(2.3) \begin{equation} ROA_t = \sum_{a\in\mathcal{A}_{t-1}}ROA_t^a + F_{t-1}C_{t-1}\end{equation}

since we have fixed yearly time steps and $F_{t-1}$ is the corresponding forward rate. Combining (2.2) with (2.1) yields the implication for the portfolio that

(2.4) \begin{align} ROA_t &= E[ROA_t|\mathcal{F}_{t-1}] + ROA_t - E[ROA_t|\mathcal{F}_{t-1}] \nonumber \\[2pt] & = \sum_{a\in\mathcal{A}_{t-1}}\Big( E[cf_t^a|\mathcal{F}_{t-1}] + E[\Delta BV_t^a|\mathcal{F}_{t-1}] \Big) \nonumber\\[2pt] & \quad + F_{t-1}C_{t-1} + ROA_t - E[ROA_t|\mathcal{F}_{t-1}] \nonumber \\[2pt] \notag &= \sum_{a\in\mathcal{A}_{t-1}}\Big( (1+F_{t-1})MV_{t-1}^a - E[MV_t^a|\mathcal{F}_{t-1}] + E[\Delta BV_t^a|\mathcal{F}_{t-1}] \Big) \nonumber \\[2pt] & \quad + F_{t-1}C_{t-1} + ROA_t - E[ROA_t|\mathcal{F}_{t-1}] \nonumber \\[2pt] &= F_{t-1}BV_{t-1} + F_{t-1}UG_{t-1} - E[{\Delta}UG_t|\mathcal{F}_{t-1}] + ROA_t - E[ROA_t|\mathcal{F}_{t-1}], \end{align}

where we have used that $\sum_{a\in\mathcal{A}_{t-1}} E[UG_t^a|\mathcal{F}_{t-1}] = E[UG_t|\mathcal{F}_{t-1}]$ . The significance of this equation is that it splits $ROA_t$ into three terms: the forward yield on the total book value, $F_{t-1}BV_{t-1}$ ; a term depending on the $\mathcal{F}_{t-1}$ -prediction of return due to realizations of unrealized gains, $F_{t-1}UG_{t-1} - E[{\Delta} UG_t|\mathcal{F}_{t-1}]$ ; finally, the difference between return and $\mathcal{F}_{t-1}$ -predicted return. This interpretation is illustrated in the following toy example.

The market value of the portfolio at t is then $MV_t = (1+F_t)^{-1}(1+K)N$ . At $t-1$ , assume that $K\ge F_{t-1}$ and $E[(1+F_t)^{-1}|\mathcal{F}_{t-1}](1+K)\ge1$ such that

\[MV_{t-1}= (1+F_{t-1})^{-1}E[(1+F_t)^{-1}|\mathcal{F}_{t-1}](1+K)N + (1+F_{t-1})^{-1}KN\ge N,\]

whence $BV_{t-1}=\text{min}(N,MV_{t-1}) = N$ . The formula for $MV_{t-1}$ is, in fact, independent of management decisions at t which reflects the fact that future investment strategies cannot affect the currently given market consistent value of a portfolio. Now we describe the constituents of Equation (2.4) under two management decisions at t: decision (1) is defined as the management’s default strategy of not taking any action at t, while decision (2) shall mean that management decides to sell one of the two bonds at t, namely $a_1$ . For example, this decision might depend on the observation of $F_t$ .

In both cases, we clearly have $F_{t-1}BV_{t-1} = F_{t-1}N$ , independently of management actions or market movements in $F_t$ .

Decision (1): using (2.1), the second term in (2.4) is now calculated as

\begin{align*}F_{t-1}UG_{t-1} - \Delta E[UG_t|\mathcal{F}_{t-1}]&= F_{t-1}(MV_{t-1} - N) - E[MV_{t}|\mathcal{F}_{t-1}] + MV_{t-1} \nonumber \\[5pt] & \quad + E[\text{min}(MV_t,N)|\mathcal{F}_{t-1}] - N \\[5pt]&= (K-F_{t-1})N \\[5pt]& \quad - E[\text{max}((F_t - K)/(1+F_t),0) | \mathcal{F}_{t-1}] N .\end{align*}

The third term gives

\begin{align*} ROA_t - E[ROA_t|\mathcal{F}_{t-1}] &= cf_t^{a_1} + cf_t^{a_2} + \text{min}((1+F_t)^{-1}(1+K)N, N) - N - KN \\[5pt] & \quad - E[\text{min}((1+F_t)^{-1}(1+K)N, N)|\mathcal{F}_{t-1}] + N \\[5pt] &= \text{min}((1+F_t)^{-1}(1+K)N, N) \\[5pt]&\quad - E[\text{min}((1+F_t)^{-1}(1+K)N, N)|\mathcal{F}_{t-1}]\end{align*}

since $cf_t^{a_1} = KN/2$ with respect to strategy (1) and $cf_t^{a_2} = KN/2$ in all cases.

Decision (2): the second term remains unchanged since it corresponds to an $\mathcal{F}_{t-1}$ prediction. This term therefore captures the contribution of unrealized gains to book value return, $ROA_t$ , that is due to the expected evolution of the portfolio. On the other hand, the third term becomes now, due to $cf_t^{a_1} = KN/2 + MV_t^{a_1}$ (coupon payments reward for having held the asset over the period $[t-1,t]$ and thus precede market placements) and $BV_t^{a_1} = 0$ (termination of asset), to

\begin{align*} ROA_t - E[ROA_t|\mathcal{F}_{t-1}] &= MV_t^{a_1} + \text{min}((1+F_t)^{-1}(1+K)N, N)/2 \\[5pt] & \quad - E[\text{min}((1+F_t)^{-1}(1+K)N, N)|\mathcal{F}_{t-1}] \\[5pt] &= (1+F_t)^{-1}(1+K)N / 2 \\[5pt] & \quad + \text{min}((1+F_t)^{-1}(1+K)N, N)/2 \\[5pt] & \quad - E[\text{min}((1+F_t)^{-1}(1+K)N, N)|\mathcal{F}_{t-1}] .\end{align*}

Notice that $MV_t^{a_1} =(BV_t^{a_1})' + (UG_t^{a_1})'$ with $(BV_t^{a_1})'= \text{min}((1+F_t)^{-1}(1+K)N, N)/2$ where $(\!\cdot\!)'$ denotes the asset’s value before selling. Hence, this corresponds to a realization of unrealized gains, $(UG_t^{a_1})' = \text{max}( (K-F_t)/$ $(1+F_t), 0 )N/2$ , as a cash flow. If $K\le F_t,$ it follows that $(UG_t^{a_1})' = 0$ , and in this case the decision to sell does not have any effect on the return because of the lower of market value or cost principle. The third term thus represents the unpredicted return (due to market movements or management actions at t).

Thus, this remark illustrates how market fluctuations and management decisions at t influence the third term, $ROA_t - E[ROA_t|\mathcal{F}_{t-1}]$ , in (2.4) while leaving the other two terms unaffected. The above formulae can be simplified by considering the special cases $F_t = 0$ or $K=0$ with $F_t>0$ .

2.2. Contracts and model points

We now describe the contracts which exist in the company’s liability portfolio at valuation time. All contracts are assumed to have a minimum guaranteed benefit and bonus benefit which depends on the company’s profit and is shared between policyholder, shareholder and tax office. Each contract gives rise to either a single maturity or mortality benefit, or multiple annuity benefits, or a surrender benefit, cost payments associated with the contract and premium payments accepted by the company. We refer to Gerber (Reference Gerber1997) for a detailed exposition of life insurance mathematics.

Each contract, c, has at each time step, t, a mathematical reserve, $\widetilde{V}_t^c$ , defined by actuarial principles and first order assumptions. Further, at valuation time the contract may have already received bonus declarations. The sum of these declared bonuses up to and including time $t=0$ are denoted by $(\widetilde{DB}_0^{\le0})^c$ . If the contract is still active at $t>0,$ this account remains unchanged, $(\widetilde{DB}_t^{\le0})^c=(\widetilde{DB}_0^{\le0})^c$ . The sum of bonus declarations after valuation time up to and including time $t>0$ is denoted by $\widetilde{DB}_t^c$ , and we have $\widetilde{DB}_0^c = 0$ . The contract total reserve is thus $\widetilde{LP}_t^c = \widetilde{V}_t^c + (\widetilde{DB}_t^{\le0})^c + \widetilde{DB}_t^c$ .

If $m>0$ is the contract’s maturity, the policyholder receives a guaranteed minimum benefit, $gbf_m^c$ , plus declared bonuses, that is, the benefit cash flow at m equals $gbf_m^c + (\widetilde{DB}_{m-1}^{\le0})^c + \widetilde{DB}_{m-1}^c$ . Profit that is generated at m is not shared with contracts maturing at the same time. Annuity payments are analogous with a corresponding fraction of bonuses being paid out. If the policyholder dies at $0<t<m,$ the death benefit similarly consists of a sum of a guaranteed minimum benefit, $gbf_t^c$ , and declared bonuses, $(\widetilde{DB}_{t-1}^{\le0})^c + \widetilde{DB}_{t-1}^c$ . Notice that, contrary to $\widetilde{DB}_{t-1}^c$ , the cash flow due to $(\widetilde{DB}_{t-1}^{\le0})^c$ is already guaranteed at valuation time since these bonuses have already been declared. If the policyholder decides to surrender the contract at $0<t<m,$ the resulting surrender benefit is $(1 - \chi_t^c)\widetilde{LP}_{t-1}^c$ where $0\le\chi_t^c\le1$ is the fraction which gives rise to the surrender gain made by the company.

For modeling purposes, it is advantageous to describe model points instead of contracts. A model point is defined as either a single contract, or a collection of identical contracts, such that survival probabilities are already taken into account. Thus, if x is the model point associated with a contract c and $p_0^t$ is the survival probability (incorporating mortality and surrender) between 0 and t the reserves are related by $LP_t^x = p_0^t\widetilde{LP}_t^c$ . However, $p_0^t$ may depend on the underlying economic scenario via dynamic policyholder behavior and therefore we refrain from introducing these probabilities explicitly. Rather, we let $V_t^x$ , $(DB_t^{\le0})^x$ , and $DB_t^x$ denote the mathematical reserve, declared bonuses up to and including valuation time, and declared bonuses after valuation time, respectively, such that survival probabilities are taken into account. If policyholder behavior is dynamic with respect to economic scenarios, then $V_t^x$ and $(DB_t^{\le0})^x$ are also scenario-dependent. Future declared bonuses, $DB_t^x$ , depend on economic scenarios by construction. Notice that $(DB_t^{\le0})^x\le (DB_0^{\le0})^x$ , in general. Analogously, the probability weighted minimum guaranteed (maturity and mortality) benefits generated by model point x at t are denoted by $gbf_t^x$ . The probability weighted cash flows, at t, due to $(DB_{t-1}^{\le0})^x$ are called $(gbf_t^{\le0})^x$ . Those due to $DB_{t-1}^x$ are called $ph_t^x$ .

The total benefit cash flows are

\[ gbf_t = \sum_{x\in\mathcal{X}_t}gbf_t^x, \qquad gbf_t^{\le0} = \sum_{x\in\mathcal{X}_t}\left(gbf_t^{\le0}\right)^x, \qquad ph_t = \sum_{x\in\mathcal{X}_t}ph_t^x\]

and the total reserves are given correspondingly by

\[ V_t = \sum_{x\in\mathcal{X}_t}V_t^x, \qquad DB_t^{\le0} = \sum_{x\in\mathcal{X}_t}\left(DB_t^{\le0}\right)^x, \qquad DB_t = \sum_{x\in\mathcal{X}_t}DB_t^x,\]

where $\mathcal{X}_t$ denotes the set of model points active at time t.

2.3. Gross surplus and profit sharing

Let $L_t$ denote the book value of liabilities at time t, and we assume that $L_t =LP_t + SF_t$ is a sum of two items: firstly, the life assurance provision, $LP_t = V_t + DB_t^{\le0} + DB_t$ ; and secondly, the surplus fund, $SF_t$ . The surplus fund at time t, $SF_{t}$ , consists of those profits that have not yet been declared to policyholders. As opposed to $LP_t$ , $SF_t$ belongs to the collective of policyholders and cannot be attributed to individual contracts. This setup follows the same logic as Gerstner et al. (Reference Gerstner, Griebel, Holtz, Goschnick and Haep2008) where $V_t$ , $DB_t^{\le0}+DB_t$ , and $SF_t$ are referred to as the actuarial reserve, allocated bonus, and free reserve (buffer account), respectively.

The difference between total book value of assets and liabilities is the free capital, $ BV_t = FC_t + L_t$ . Since return on free capital is not shared with policyholders and therefore does not contribute to the future discretionary benefits that we are interested in, we assume without loss of generality that $FC_t = 0$ , so that the initial book value of assets is equal to the initial value of liabilities. Further, we assume that all shareholder gains that are produced by the company over the projection time are directly paid out to shareholder and not accumulated within the company so that $FC_t=0$ (cf. Hochgerner and Gach, Reference Hochgerner and Gach2019, A. 2.2):

(2.5) \begin{equation} BV_t = L_t = LP_t+SF_t.\end{equation}

Indeed, for purposes of best estimate calculation the free capital is not relevant as this and the corresponding revenue is not shared with policyholders. Moreover, assets used to cover statutory reserves may not be attributed separately to $L_t$ and $FC_t$ (Bundesministerium der Finanzen (BMF); Verordnung). Hence, setting $FC_t=0$ leads to the appropriate scaling of revenue that is to be shared with policyholders. Thus, the equality (2.5) can always be achieved by replacing $BV_0$ , $MV_0$ , $UG_0$ by $BV'_{\!\!0} = BV_0-FC_0$ , $MV'_{\!\!0} = \frac{BV'_{\!\!0}}{BV_0}MV_0$ , $UG'_{\!\!0} = \frac{BV'_{\!\!0}}{BV_0}UG_0$ , respectively, and assuming that future shareholder gains are paid out as cash flows which leave the model. On the other hand, if the goal is to build a comprehensive asset liability model to simulate the shareholder’s point of view then the free capital is of paramount importance. However, in the simulation of a run-off liability book as under Solvency II such a point of view is difficult to realize since the relation between $FC_t$ and $L_t$ quickly becomes unrealistic (without introducing new business). Notice also that the equity position in the balance sheet model of Gerstner et al. (Reference Gerstner, Griebel, Holtz, Goschnick and Haep2008) is a hybrid of free capital, in the above sense of $FC_t$ , and hidden reserves, $UG_t$ . We do retain $UG_t$ in the projection since this is indispensable for best estimate calculation.

The profit sharing mechanism depends on the company’s gross surplus with respect to local accounting rules. The gross surplus can be described verbally as the sum of book value return, increase or decrease of statutory reserves, and all relevant cash flows (premiums, benefits, costs). The gross surplus, $gs_t$ , at t is therefore defined as

\begin{align*} gs_t & \;:\!=\; ROA_t - \Delta V_t - \Delta DB_t^{\le0} - DB_t^- + DB_{t-1} \\[5pt] & \;\;\quad + pr_t - gbf_t - gbf_t^{\le0} - ph_t - co_t,\end{align*}

where $ROA_t$ is the book value return (2.4), $DB_t^-$ is the account of declared bonuses before bonus declaration at t, $pr_t$ are premium payments and $co_t$ are all cost cash flows. If $\chi_t$ is the appropriately averaged surrender fee factor at t, the discrepancy between $- DB_t^- + DB_{t-1}$ and $ph_t$ can be expressed as $- DB_t^- + DB_{t-1} - ph_t = \chi_t DB_{t-1}$ . The analogous expression holds with an appropriately chosen factor $\chi_t^{\le0}$ , representing surrender fees: $-\Delta DB_t^{\le0} = \chi_t^{\le0} DB_{t-1}^{\le0}$ . Let further $\rho_t$ denote the averaged technical interest rate at $t-1$ such that $-\Delta V_t + pr_t - gbf_t - co_t = -\rho_t V_{t-1} {-} tg_t$ where $tg_t$ is the technical gains due to mortality and cost margins and surrender fees stemming from the mathematical reserves. We define $\gamma_t \;:\!=\; (tg_t + \chi_t^{\le0} DB_{t-1}^{\le0} + \chi_t DB_{t-1})/LP_{t-1}$ and express the gross surplus as

(2.6) \begin{equation} gs_t = ROA_t - \rho_t V_{t-1} + \gamma_t LP_{t-1}.\end{equation}

If $gs_t$ is positive, it is the surplus shared between policyholder, shareholder and tax office. If it is negative, it is covered by the shareholder. Indeed, profit sharing is defined by legislation (Bundesministerium der Finanzen (BMF); Verordnung; Dorobantu et al., Reference Dorobantu, Salhi and Therond2020) and requires that $gs_t$ is shared between shareholders, policyholders and tax office according to

(2.7) \begin{equation} gs_t = sh_t + ph_t^* + tax_t,\end{equation}

where $sh_t = gsh\cdot gs_t^+ - gs_t^-$ , $ph_t^* = gph\cdot gs_t^+$ and $tax_t = gtax\cdot gs_t^+$ , and where gsh, gph and gtax are positive numbers such that $gsh + gph + gtax = 1$ . Furthermore, $c^+$ and $c^-$ denote the positive and negative parts of a number c, respectively.

Notice that $sh_t$ and $tax_t$ constitute cash flows since this is money that leaves the company (and the model), while $ph_t^*$ is an accounting flow since this corresponds to a quantity that is transferred within the company to a different account but is not paid out as a benefit at time t.

A fundamental principle of traditional life insurance is that profit sharing is not equal to profit declaration (Bundesministerium der Finanzen (BMF); Verordnung). This means that $ph_t^* = gph\cdot gs_t^+$ is not necessarily declared (or credited) to its full extent to specific policyholder accounts. Rather, the profit sharing mechanism in traditional life insurance is such that a part, $\nu_t\cdot ph_t^*$ with $0\le\nu_t\le1$ , of $ph_t^*$ is declared to the policyholder accounts. Management may choose the value $\nu_t$ at each accounting step t. However, declaration may also be augmented by additional contributions, $\eta_t\cdot SF_{t-1}$ with $0\le\eta_t\le1$ , from the previously existing surplus fund $SF_{t-1}$ . Again, management may choose the value $\eta_t$ at each accounting step t. Typically, surplus fund contributions will take place when $ph_t^*$ is small compared to management goals.

The total bonus declaration to $DB_t$ at time t is therefore of the form

(2.8) \begin{equation} \nu_t\cdot ph_t^* + \eta_t\cdot SF_{t-1}\end{equation}

with the factor $\nu_t$ and $\eta_t$ determined according to management rules.

As above, let $ph_t$ denote the amount of discretionary benefits paid out at time t. This cash flow depends on declarations to the declared benefits account, $DB_k$ , which have occurred at times $0<k<t$ . Declarations at valuation time, $t=0$ , belong by definition to $DB_t^{\le0}$ , and the resulting cash flows are already guaranteed at $t=0$ , whence these do not contribute to $ph_t$ . Therefore, we have

(2.9) \begin{equation} ph_1=0.\end{equation}

For $k>0,$ let $0\le\eta_k\le1$ denote the amount of declaration from $SF_{k-1}$ to $DB_k$ . The numbers $\eta_k$ and $\nu_k$ are, in general, unknown at $t=0$ and depend on management rules. Let further $0\le \mu_k^t\le1$ denote the fraction of bonus declarations at k, $\eta_k\cdot SF_{k-1} + \nu_k\cdot ph_k^*$ , that is either paid out as a future discretionary benefit (in case of contract maturity or mortality) or kept by the company as a surrender fee (in case of premature contract termination), $sg_t^*$ , at time t. This fraction is also unknown and may depend, in general, on dynamic policyholder behavior. The defining relation is thus

(2.10) \begin{equation} ph_t + sg_t^* = \sum_{k=1}^{t-1} \mu_k^t\Big(\eta_k\cdot SF_{k-1} + \nu_k\cdot ph_k^*\Big),\end{equation}

where $t\ge2$ . Since the sum of discretionary benefits and surrender fees cannot exceed the amount of previous declarations, we must have

(2.11) \begin{equation} \sum_{t=k+1}^T\mu_k^t \le 1 .\end{equation}

To sum up the above discussion, passing from $t-1$ to t, $DB_{t-1}$ is:

• increased by declarations $\eta_t\cdot SF_{t-1}$ from the surplus fund where $0\le\eta_t\le1$ is chosen by the management,

• increased by direct policyholder declarations $\nu_t\cdot ph_t^*$ where $0\le\nu_t\le1$ is chosen by the management,

• decreased by cash flows, $ph_t$ , to policyholders whose contracts terminate at t, and

• decreased by accounting flows $sg_t^* \;:\!=\; \chi_t\cdot DB_{t-1}$ with $0\le\chi_t\le1$ due to surrender fees. The fraction $\chi_t\cdot DB_{t-1}$ is freed up, in the sense that it is not attributed to specific contracts anymore, and thus contributes to the annual gross surplus.

Therefore, we have the iterative relation

(2.12) \begin{equation} DB_t = DB_{t-1} + \eta_t\cdot SF_{t-1} + \nu_t\cdot ph_t^* - ph_t - sg_t^*\end{equation}

with starting point $DB_0 = 0$ .

Consider the surplus fund $SF_{t-1}$ at $t-1$ . Going one time step further, it is increased by allocating $ (1-\nu_t)\cdot ph_t^*$ to the fund, which is the part of $ph^*_t$ not declared to the policyholders’ accounts, and decreased by declaring $ \eta_t\cdot SF_{t-1}$ to policyholder accounts. We thus obtain

(2.13) \begin{equation} SF_t = SF_{t-1} + (1-\nu_t)\cdot ph_t^* - \eta_t\cdot SF_{t-1}\end{equation}

with known starting point $SF_0$ . Together with (2.12) this yields

(2.14) \begin{equation} \Delta\,(DB_t+SF_t) = DB_t + SF_t - DB_{t-1} - SF_{t-1} = + ph_t^* - ph_t - sg_t^*.\end{equation}

Crucially, the model-dependent fractions $\nu_t$ and $\eta_t$ do not appear in this equation. This evolution equation for $DB_t + SF_t$ is the starting point for the subsequent analysis. The sum $DB_t+SF_t$ is increased at each time step by the total shared profit, $ph_t^*$ , as opposed to the declaration (2.8), and therefore represents the statutory reserves of previously shared profit, although the totality of this sum is not a (single) balance sheet item.

For further reference, we observe that

(2.15) \begin{align} PH^* &\;:\!=\; E\Big[\sum B_t^{-1} ph_t^*\Big] = gph\cdot E\Big[\sum B_t^{-1} gs_t^+\Big] \\[5pt] &= gph\cdot E\Big[\sum B_t^{-1} gs_t\Big] + gph\cdot E\Big[\sum B_t^{-1} gs_t^-\Big]\nonumber \\[5pt] &= gph\cdot (VIF + PH^* + TAX ) + gph\cdot COG. \nonumber \end{align}

Here we have used the splitting $gs_t = sh_t + ph_t^* + tax_t$ , where $sh_t$ and $tax_t$ are the shareholder and tax cash flows as defined above, to obtain the value of in-force business

\[ VIF = E\left[\sum_{t=1}^T B_t^{-1} sh_t\right] = E\left[\sum_{t=1}^T B_t^{-1} \left(gsh\cdot gs_t^+ - gs_t^-\right) \right],\]

the cost of guarantees

(2.16) \begin{equation} COG = E\left[\sum_{t=1}^T B_t^{-1} gs_t^-\right],\end{equation}

and

\[TAX = E\left[\sum_{t=1}^T B_t^{-1} tax_t \right] = E\left[\sum_{t=1}^T B_t^{-1} gtax\cdot gs_t^+ \right].\]

2.4. Future discretionary benefits

According to Solvency II (Directive, 2009; Commission, 2014), the best estimate is the expectation of all future cash flows which are related to existing business. These cash flows are benefits, $gbf_t+gbf_t^{\le0}+ph_t$ , premium income, $pr_t$ , and costs, $co_t$ . The best estimate is thus defined as

(2.17) \begin{equation}BE \;:\!=\; E\left[\sum_{t=1}^TB_t^{-1}\left(gbf_t+gbf_t^{\le0}+ph_t + co_t - pr_t\right)\right].\end{equation}

The value of future discretionary benefits is given by

(2.18) \begin{equation}FDB \;:\!=\; E\left[\sum_{t=1}^T B_t^{-1}ph_t\right]\end{equation}

and the value of the guaranteed benefits is by definition

(2.19) \begin{equation} GB \;:\!=\; BE - FDB = E\left[\sum_{t=1}^TB_t^{-1}\left(gbf_t + gbf_t^{\le0} + co_t - pr_t\right)\right].\end{equation}

If actuarial variables are independent from economical ones, then $gbf_t + gbf_t^{\le0}$ , $co_t$ and $pr_t$ , and hence GB, may be calculated from a purely deterministic model. This independence would exclude the possibility of dynamic policyholder behavior (e. g., surrender depends dynamically on a comparison of declared bonuses and the prevailing yield curve). However, for our results to hold, we do not need to make this assumption.

Lemma 2.3.

(2.20) \begin{equation} PH^* = \frac{gph}{1-gph}\Big(MV_0 - E\left[B_T^{-1}MV_T\right] - GB - FDB + COG \Big)\end{equation}

Proof. We use the no-leakage principle (Hochgerner and Gach Reference Hochgerner and Gach2019, Prop. 2.2) which states that $MV_0 = BE + VIF + TAX + E[B_T^{-1}MV_T]$ . With the decomposition $BE=GB+FDB$ and definition (2.15), this implies that $PH^* = gph(PH^* + MV_0 - GB - FDB - E[B_T^{-1}MV_T] + COG)$ .

4.1. Liability run-off assumptions

Since the portfolio is in run-off there is a time, h, where $E[LP_h] = LP_0/2$ . Continuing from h onwards there has to be a time, $h+h'$ , such that $E[LP_{h+h^{\prime}}] = E[LP_h]/2$ . Assuming that the company’s business model has been stable over time, we have time homogeneity in the sense that $h'=h$ and run-off of the liability book is geometric. This would not be satisfied if the company under consideration has taken up business only very recently but for companies with a longer history we view this as a very good approximation.

Assumptions 4.3 and 4.4 are also statements about time homogeneity. Management rules concerning bonus declarations should remain reasonably constant in the long run such that $\sigma$ and $\vartheta$ do not vary too strongly. The relevant point in this context is that these quantities should not vary arbitrarily but follow from target rates set by management rules. Assumption 4.4 is comparable to the assumption concerning the ‘annual interest rate’ in Gerstner et al. (Reference Gerstner, Griebel, Holtz, Goschnick and Haep2008, Section 4.2).

4.2. Surrender assumption

The factor $\gamma_t$ comprises mortality, cost and surrender margins as a fraction of the full life assurance provision, $LP_{t-1}$ . It is therefore reasonable to expect that the same factor can be used as an upper bound on the surrender margin arising from declared bonuses, $DB_{t-1}$ , alone.

4.3. Bonus benefit assumptions

Because of Equation (2.13), the bonus benefit declaration at t can be expressed as $\eta_t\cdot SF_{t-1} + \nu_t\cdot ph_t^* = SF_{t-1} - SF_t + ph_t^*$ . Management rules generally strive to keep profit declarations stable while, in accordance with Assumptions 4.4 and 4.2, $SF_t$ is expected to decrease geometrically over time. In order for $SF_t$ to decrease in expectation, the bonus benefit declarations must be strictly positive in expectation. To achieve this, a fraction of the profit share, $ph_t^*$ , must also be declared to policyholders, at least in expectation. We turn this reasoning into an assumption along all scenarios.

Notice that, for fixed k, this definition entails $\sum_{s=k}^{T-1} \mu_k^{s+1} = 1-l_T^h/l_k^h = 1$ ; cf. (2.11). That is, run-off is complete at T.

4.4. Gross surplus assumptions

According to (2.6) and (2.4), the gross surplus is given by

\begin{align*} gs_t & = F_{t-1} (LP_{t-1} + SF_{t-1}) + F_{t-1}UG_{t-1} - \Delta UG_t|\mathcal{F}_{t-1} + ROA_t - ROA_t|\mathcal{F}_{t-1} \\[5pt] & \quad - (\rho_t-\gamma_t) V_{t-1},\end{align*}

where $\rho_t$ and $\gamma_t$ may, in general, also depend on the stochastic interest rate curve via dynamic surrender.

For the purpose of estimating terms III and COG in (3.2), we make the following simplifying assumptions. The principle idea behind these assumptions is that the main source of stochasticity in $gs_t$ is the forward rate $F_{t-1}$ whence all other quantities are replaced by their expected values. The simplified model of $gs_t$ will be denoted by $\widehat{gs}_t$ .

The motivation for item (2) is as follows. The quantity $F_{t-1} UG_{t-1} - {E[\Delta UG_t|\mathcal{F}_{t-1}]} =: cf_t^{UG}$ may be viewed as a cash flow due to realizations of unrealized gains as assets approach their maturities (an example of this reasoning is contained in Remark 2.1). Indeed, if an asset a has maturity $T_a$ , then we must have $UG_{T_a}^a = MV_{T_a}^a - BV_{T_a}^a = 0$ , and thus, $UG_t^a$ tends to 0 as t approaches $T_a$ . As a proxy for the number d, we take the duration of the portfolio. Assuming that cash flows, $cf_t^{UG}$ , due to realizations of unrealized gains are known at valuation time $t=0$ , we obtain $UG_0 = \sum_{t=1}^T P(0,t)cf_t^{UG}$ . Setting $\sum_{t=1}^T P(0,t)(F_{t-1} UG_{t-1} - {E[ \Delta UG_t|\mathcal{F}_{t-1}])} =\sum_{t=1}^T P(0,t)cf_t^{UG} = UG_0 = \sum_{t=1}^T(l_{t-1}^d - l_t^d)UG_0$ and insisting on equality of the summands leads to the above assumption.

This assumption reflects the general principle that book values are expected to be more stable than market values since not all market movements are reflected in book values but rather lead to unrealized gains or losses (Dorobantu et al., Reference Dorobantu, Salhi and Therond2020).

Invoking the above Assumptions 4.2, 4.9, 4.10, 4.8, 4.3 and 4.4, we define

(4.1) \begin{align} \widehat{gs}_t &\;:\!=\; F_{t-1} E[BV_{t-1}] + P(0,t)^{-1}(l_{t-1}^d - l_t^d)UG_0 - \rho_t V_{t-1} + \gamma_t LP_{t-1} \\[5pt] & = \left( F_{t-1} + P(0,t)^{-1}\frac{l_{t-1}^d-l_t^d}{l_{t-1}^h}\frac{UG_0}{(1+\vartheta)LP_0} -\frac{(1-\sigma)\rho_t-\gamma_t}{1+\vartheta} \right) (1+\vartheta)l_{t-1}^h LP_0 \nonumber \end{align}

to be used as a simplified model for $gs_t$ .

5.1. Estimating I

In accordance with Assumption 4.1, we estimate I by

(5.1) \begin{equation} \widehat{I} = 0.\end{equation}

Compare also with the second statement in Hochgerner and Gach (Reference Hochgerner and Gach2019, Prop. 2.2).

5.2. Estimating II

The expression $sg_t^* = \chi_{t} DB_{t}$ in Term II corresponds to the fraction of $DB_{t}$ that is freed up each year due to policyholder surrender fees and thus contributes to the company’s surplus as a component of the surrender gains. Assumptions 4.10, 4.5, 4.3 and 4.2 imply that term II can be estimated as $II\le\widehat{II}$ with

(5.2) \begin{align} \widehat{II} \;:\!=\; (1-gph)\sum_{t=2}^T\gamma_t\sigma_t P(0,t)l_{t-1}^h LP_0.\end{align}

5.3. Bounding III from above

The essential idea is to use the recursive relation (2.14) to obtain an upper bound for $DB_t+SF_t$ . Equation (2.14) implies

(5.3) \begin{align} DB_t + SF_t &= SF_0 + \sum_{s=1}^t ph_s^* - \sum_{s=2}^t (ph_s+sg_s^*) = SF_0 + gph\cdot gs_t^+ \nonumber \\[5pt] & \quad + \sum_{s=1}^{t-1}\Big( gph\cdot gs_s^+ - ph_{s+1} - sg_{s+1}^* \Big).\end{align}

Assumptions 4.6 and 4.7 yield

\begin{align*} \sum_{s=1}^{t-1}\Big(ph_{s+1}+sg_{s+1}^*\Big) &= \sum_{s=1}^{t-1}\sum_{k=1}^s\mu_k^{s+1}\Big(\eta_k\cdot SF_{k-1} + \nu_k\cdot ph_k^*\Big)\\ &\ge \nu \sum_{s=1}^{t-1}\sum_{k=1}^s\mu_k^{s+1}ph_k^* = \nu\sum_{k=1}^{t-1}\sum_{s=k}^{t-1}\frac{l_s^h-l_{s+1}^h}{l_k^h}ph_k^* \\ & = \nu\sum_{k=1}^{t-1}\frac{l_k^h-l_{t}^h}{l_k^h}ph_k^*,\end{align*}

whence (5.3) satisfies

(5.4) \begin{equation} DB_t+SF_t \le SF_0 + gph\cdot gs_t^+ + gph\cdot \sum_{s=1}^{t-1} \Big( 1 - \nu (1 - l_{t-s}^h ) \Big) gs_s^+ .\end{equation}

Thus,

(5.5) \begin{align} \!\!\!III &= (1-gph)\sum_{t=0}^{T-1} E\Big[ B_{t+1}^{-1}F_{t}(DB_{t}+SF_{t}) \Big] \\ & \quad \!\le (1-gph) (1-P(0,T) ) SF_0 + (1-gph)gph \sum_{t=1}^{T-1} E\Big[B_{t+1}^{-1}F_{t} \cdot gs_t^+ \Big] \nonumber \\ & \!\quad + (1-gph)gph \sum_{t=2}^{T-1} \sum_{s=1}^{t-1} \Big( 1 - \nu (1 - l_{t-s}^h ) \Big) E\Big[B_{t+1}^{-1}F_{t} \cdot gs_s^+ \Big]. \nonumber \end{align}

The expression $E[B_{t+1}^{-1}F_{t} \cdot gs_s^+ ]$ gives the fair value of the risk free return on $gs_s^+$ in the period from t to $t+1$ .

5.4. Bounding III from below

We use again Equation (5.3) and notice that Equations (2.10) and (2.13) imply

\begin{align*} \sum_{s=2}^t\Big( ph_s + sg_s^*\Big) &= \sum_{s=2}^t \sum_{k=1}^{s-1} \mu_k^s\Big(SF_{k-1} - SF_k + ph_k^*\Big) \\ & = \sum_{k=1}^{t-1} \sum_{s=k+1}^t \mu_k^s \Big( SF_{k-1} - SF_k + ph_k^* \Big) \\ &\le SF_0 - SF_{t-1} + \sum_{k=1}^{t-1}ph_k^*\end{align*}

since $\sum_{s=k+1}^t \mu_k^s\le 1$ due to (2.11). Inserting this in (5.3) yields $DB_t + SF_t \ge ph_t^* + SF_{t-1}$ for all $t\ge1$ , and therefore

\begin{align*} III &= (1-gph)E\left[\sum_{t=0}^{T-1} F_t B_{t+1}^{-1}(DB_t+SF_t)\right] \\[5pt] \notag & \quad \ge (1-gph) F_0(1+F_0)^{-1} SF_0 \\ & \quad + (1-gph)E\left[\sum_{t=1}^{T-1} F_t B_{t+1}^{-1} \Big( gph\cdot gs_t^+ + SF_{t-1} \Big)\right] .\end{align*}

This estimate does not depend on any of the assumptions in Section 4. Neglecting, in accordance with Assumption 4.10, the variation of $SF_{t-1}$ in comparison to that of $F_{t-1}$ , and using Assumptions 4.4 and 4.2, yields

(5.6) \begin{align} III & \ge (1-gph)\left( F_0(1+F_0)^{-1} SF_0 + \vartheta\sum_{t=1}^{T-1}(P(0,t)-P(0,t+1) )l_{t-1}^h LP_0 \right) \nonumber \\[5pt] & + gph(1-gph)E\Big[\sum_{t=1}^{T-1} F_t B_{t+1}^{-1} gs_t^+ \Big] .\end{align}

5.5. Estimating the return on the deferred caplet

Let us rewrite $F_{t}B_{t+1}^{-1} gs_s^+= (B_{t}^{-1}-B_{t+1}^{-1}) gs_s^+= (D(s,t)-D(s,t+1)) B_s^{-1} gs_s^+$ and abbreviate the coefficients of variations as

(5.7) \begin{equation} CV_{s,t}^1 \;:\!=\; CV\Big[D(s,t)-D(s,t+1)\Big] ,\qquad CV_s^2 \;:\!=\; CV\Big[ B_s^{-1} gs_s^+\Big] .\end{equation}

Since $-1 \le Corr_s^t \;:\!=\; Corr[D(s,t)-D(s,t+1), B_s^{-1} gs_s^+] \le 1$ , it follows that

\begin{align*} E\Big[(D(s,t&)-D(s,t+1)) B_s^{-1} gs_s^+ \Big] = E\Big[(D(s,t)-D(s,t+1))\Big] E\Big[ B_s^{-1} gs_s^+ \Big] \\[5pt] &\phantom{===} + Corr_s^t \cdot CV_{s,t}^1 CV_s^2 E\Big[(D(s,t)-D(s,t+1))\Big] E\Big[ B_s^{-1} gs_s^+ \Big]\\[5pt] &\le \Big(P(s,t) - P(s,t+1)\Big) E\Big[ B_s^{-1} gs_s^+ \Big] \Big( 1 + CV_{s,t}^1 CV_s^2 \Big)\end{align*}

and

\begin{align*} & E\Big[(D(s,t)-D(s,t+1)) B_s^{-1} gs_s^+ \Big] \\[5pt] & \ge \Big(P(s,t) - P(s,t+1)\Big) E\Big[ B_s^{-1} gs_s^+ \Big] \Big( 1 - CV_{s,t}^1 CV_s^2 \Big).\end{align*}

Hence, (5.6) and (5.5) lead to

(5.8) \begin{align} &\;(1-gph)\left( F_0(1+F_0)^{-1} SF_0 + \vartheta\sum_{t=1}^{T-1}(P(0,t)-P(0,t+1) )l_{t-1}^h LP_0 \right) \nonumber \\[5pt] & \qquad + gph(1-gph)\sum_{t=1}^{T-1} \Big(1 - CV_{0,t}^1\, CV_t^2 \Big) \Big(1 - P(t,t+1)\Big) E\Big[ B_{t}^{-1} gs_t^+ \Big] \nonumber\\[5pt] &\le III \\[5pt] &\le (1-gph) (1-P(0,T)) SF_0 \nonumber \end{align}

\begin{align*} & \qquad + (1-gph)gph \sum_{t=1}^{T-1} \Big(1 + CV_{0,t}^1\, CV_t^2 \Big) \Big( 1 - P(t, t+1) \Big) E\Big[B_{t}^{-1} gs_t^+ \Big] \nonumber\\[5pt] &\qquad + (1-gph)gph \sum_{t=2}^{T-1} \sum_{s=1}^{t-1} \Big( 1 - \nu (1 - l_{t-s}^h ) \Big) \Big(1 + CV_{s,t}^1\, CV_s^2 \Big) \nonumber\\[5pt] & \;\;\qquad \Big(P(s,t) - P(s,t+1)\Big) E\Big[ B_s^{-1} gs_s^+ \Big]. \nonumber\end{align*}

The term $E[ B_s^{-1} gs_s^+ ]$ is the value of the caplet with payoff $gs_s^+$ at time s.

5.6. Estimating the caplet

Now we replace $gs_s$ by its simplified model $\widehat{gs}_s$ defined in (4.1). This implies that the (simplified) caplet can be expressed as

(5.9) \begin{equation} E\Big[ B_s^{-1} \widehat{gs}_s^+ \Big] = \mathcal{O}_s^+ (1+\vartheta)l_{s-1}^h LP_0,\end{equation}

where

(5.10) \begin{equation} \mathcal{O}_s^{\pm} \;:\!=\; E\left[B_s^{-1}\Big( F_{s-1} + P(0,s)^{-1}\frac{l_{s-1}^d-l_s^d}{l_{s-1}^h}\frac{UG_0}{(1+\vartheta)LP_0} - \frac{(1-\sigma)\rho_s-\gamma_s}{1+\vartheta} \Big)^{\pm}\right]\end{equation}

is the value at 0 of the caplet (corresponding to $+$ ) or floorlet (corresponding to $-$ ) with maturity $s-1$ and payment $( F_{s-1} + P(0,s)\frac{l_{s-1}^d-l_s^d}{l_{s-1}^h}\frac{UG_0}{(1+\vartheta)LP_0} - \frac{(1-\sigma)\rho_s-\gamma_s}{1+\vartheta} )^{\pm}$ occurring at settlement date s. In the normal model, this value is given by the Black formula (Black, Reference Black1976; Brigo and Mercurio, Reference Brigo and Mercurio2006)

(5.11) \begin{equation} \mathcal{O}_s^{\pm} = P(0,s)\cdot\Big( \pm ( F_{s-1}^0 - k_s )\Phi(\pm \kappa_s) + \text{IV}_s \sqrt{s}\phi(\pm \kappa_s) \Big),\end{equation}

where $\Phi$ and $\phi$ are the normal cumulative distribution and density functions, respectively. Further,

\[ \kappa_s \;:\!=\; \frac{ F_{s-1}^0 - k_s } { \text{IV}_s\sqrt{s} },\]

where $F_{s-1}^0 = P(0,s-1) - P(0,s)$ is the forward rate prevailing at time 0, the strike is given by

(5.12) \begin{equation} k_s \;:\!=\; - P(0,s)^{-1}\frac{l_{s-1}^d-l_s^d}{l_{s-1}^h}\frac{UG_0}{(1+\vartheta)LP_0} + \frac{(1-\sigma)\rho_s-\gamma_s}{1+\vartheta}\end{equation}

and $\text{IV}_s$ is the caplet implied volatility known from market data. Using the normal model at this point is, of course, an additional model choice.

Therefore, estimate (5.8) may be reformulated as

(5.13) \begin{equation} \widehat{III}_{lb} \le III \le \widehat{III}_{ub}\end{equation}

with

(5.14) \begin{align} \widehat{III}_{lb} &\;:\!=\; (1-gph)\left( F_0(1+F_0)^{-1} SF_0 + \vartheta\sum_{t=1}^{T-1}(P(0,t)-P(0,t+1) )l_{t-1}^h LP_0 \right) \nonumber \\ & \quad + gph(1-gph)\sum_{t=1}^{T-1} \! \Big(1 - CV_{0,t}^1\, CV_t^2 \Big) \Big(1 - P(t,t+1)\Big) \mathcal{O}_t^+ (1+\vartheta)l_{t-1}^h LP_0 \\ \widehat{III}_{ub} &\;:\!=\; (1-gph) (1-P(0,T)) SF_0 \nonumber \\ &\quad + (1-gph)gph \sum_{t=1}^{T-1} \Big(1 + CV_{0,t}^1\, CV_t^2 \Big) \Big( 1 - P(t, t+1) \!\Big) \mathcal{O}_t^+ (1+\vartheta)l_{t-1}^h LP_0 \nonumber \\ & \quad + (1-gph)gph \sum_{t=2}^{T-1} \sum_{s=1}^{t-1} \Big( 1 - \nu (1 - l_{t-s}^h ) \Big) \Big(1 + CV_{s,t}^1\, CV_s^2 \Big) \Big(P(s,t) \nonumber \\ & \quad - P(s,t+1)\Big) \mathcal{O}_s^+ (1+\vartheta)l_{s-1}^h LP_0. \nonumber \end{align}

5.7. Approximating COG

The shareholder cost of guarantees is defined in (2.16). We use again the simplified model (4.1) to estimate COG by

(5.15) \begin{align} \widehat{COG} \;:\!=\; E\left[ \sum_{t=1}^T B_t^{-1} \widehat{gs}_t^- \right] = \sum_{t=1}^T \mathcal{O}_t^- (1+\vartheta)l_{t-1}^h LP_0,\end{align}

where $\mathcal{O}_t^-$ is the value of the floorlet given in (5.11).

5.8. Estimating FDB

Under the assumptions of Section 4, the terms COG, I, II and III can be estimated by $\widehat{COG}$ , $\widehat{I}=0$ , $II\le\widehat{II}$ and $\widehat{III}_{lb} \le III \le \widehat{III}_{ub}$ as defined by (5.15), (5.1), (5.2) and (5.14), respectively. These estimates yield a lower bound, $\widehat{LB}$ , and an upper bound, $\widehat{UB}$ , for FDB:

(5.16) \begin{equation} \widehat{LB} \le FDB \le \widehat{UB},\end{equation}

where

(5.17) \begin{align} \widehat{LB} &\;:\!=\; SF_0 + gph\Big(LP_0+UG_0-GB\Big) - \widehat{II} - \widehat{III}_{ub} \end{align}

(5.18) \begin{align} \widehat{UB} &\;:\!=\; SF_0 + gph\Big(LP_0+UG_0-GB\Big) + gph\cdot\widehat{COG} - \widehat{III}_{lb}.\end{align}

If the difference $\widehat{UB}-\widehat{LB}$ is sufficiently small (e.g., in comparison to $BV_0$ ), then $\widehat{FDB} = (\widehat{LB}+\widehat{UB})/2$ may be used as an estimator for FDB.

These estimation formulas are analytic in the sense that they do not depend on a numerical model. For the reader’s convenience, we provide a compact list of data which have to be known or estimated in order to calculate these bounds in Table 2.

Table 2 List of data needed to calculate $\widehat{LB}$ and $\widehat{UB}$ .

6.1. Allianz Lebensversicherungs-AG: publicly reported values

The data in Table 3 are taken from publicly available reports for the accounting years 2017–2019. The relevant references are listed in Table 4.

The value of $UG_0$ is already scaled to $L_0$ , which is in line with the general assumption (2.5). The reason behind this scaling is that according to Bundesministerium der Finanzen (BMF), 3 only the fraction of the capital gains, corresponding to the assets scaled to cover the average value of liabilities in the accounting year under consideration, contribute to the gross surplus.

As for $L_0$ , we adjust the local GAAP value of life insurance with profit participation for necessary regrouping of business, as explained in Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2017b, p. 52), Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2018b, p. 46), Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2019b, p. 46) for the different accounting years 2017–2019.

Table 3 Allianz Lebensversicherungs-AG: public data for 2017–2019, values are in billion euros.

Table 4 Allianz Lebensversicherungs-AG: references for the data listed in Table 3.

ⁱ Versicherung mit berschussbeteiligung.

ⁱⁱ Stille Reserven der einzubeziehenden Kapitalanlagen.

ⁱⁱⁱ Rckstellung fr Beitragsrckerstattung abzglich festgelegte, aber noch nicht zugeteilte Teile.

^iv berschussfonds.

^v Bester Schtzwert: Wert fr garantierte Leistungen.

^vi Bester Schtzwert: zuknftige berschussbeteiligung.

Table 5 BaFin: public data for 2017–2019, values are in billion euros.

ⁱ Überschuss.

ⁱⁱ Direktgutschrift.

ⁱⁱⁱ Zuführung zur RfB.

^iv Kapitalanlagenergebnis 1 b).

^v Tabelle 130, Versicherungstechnische Rückstellungen brutto, selbst abgeschlossenes Geschäft

^vi Bilanzposten a) 6 a) brutto: Tabelle 130, Versicherungstechnische Rckstellungen, soweit das Anlagerisiko vom Versicherungsnehmer getragen wird.

Table 6 Values of $\gamma$ for 2017–2019.

6.2. Estimating technical gains from market data

For the German life insurance market, technical gains can be determined from tables 130 and 141 in Bundesanstalt (2019). The relevant items are stated below:

We estimate the technical gains relative to the life assurance provisions by $\widehat{\gamma}_{LP} \;:\!=\; (a+b-d)/(e-f)$ and find

6.3. Estimating $\rho$

The average technical interest rate of the Allianz Lebensversicherungs-AG can be derived from the distribution of life assurance provision over the guaranteed interest rates, with the following results:

The underlying data can be found in Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2017a, p. 34), Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2018a, p. 33), and Allianz Lebensversicherungs-AG (Reference Allianz Lebensversicherungs-AG2019a, p. 37). (To obtain the values stated in Table 7, we have taken the upper end points where technical interest rate intervals are provided.)

Table 7 Values of $\widehat{\rho}$ for 2017–2019.

6.4. Calculating gph

The net policyholder shares, nph for the accounting years 2017–2019, can be obtained via $nph=(b+c)/a$ from the values collected in the table below (see Allianz Lebensversicherungs-AG, Reference Allianz Lebensversicherungs-AG2017a, p. 9, Allianz Lebensversicherungs-AG, Reference Allianz Lebensversicherungs-AG2018a, p. 9, and Allianz Lebensversicherungs-AG, Reference Allianz Lebensversicherungs-AG2019a, p. 8 for accounting years 2017–2019).

The gross policyholder share, gph, is calculated from nph according to the relation $gph = (1-\tau) nph / (1-\tau\cdot nph)$ . Applying the German tax rate of $\tau = 29.9\%$ (Bundesministerium der Finanzen (BMF) 2019, p. 16) yields the following table:

Table 8 Values are in billion euros.

ⁱ Bruttoüberschuss.

ⁱⁱ Zuführung zur RfB.

ⁱⁱⁱ Direktgutschrift.

For the estimation of Term III, we fix $gph = 75.5\%$ as the average of the values in Table 9.

Table 9 Values of gph for 2017–2019.

6.5. Discount rates

The following are the publicly available EIOPA discount rates for 2017, 2018 and 2019.